Big Bounce and Dark Energy in Gravity with Torsion
Nikodem J. Popławski
Lecture Virtual Institute of Astroparticle Physics Paris, France
22 June 2012 Outline
1. Problems of standard cosmology 2. Einstein-Cartan-Sciama-Kibble theory of gravity 3. Dirac spinors in spacetime with torsion 4. Solution: cosmology with torsion • Nonsingular big bounce instead of singular big bang • Torsion as simplest alternative to inflation 5. Simplest affine theory of gravity • Cosmological constant from torsion E = mc2
Problems of standard cosmology
• Big-bang singularity – can be solved by LQG But LQG has not been shown to reproduce GR in classical limit
• Flatness and horizon problems – solved by inflation consistent with cosmological perturbations observed in CMB But: - Scalar field with a specific (slow-roll) potential needed fine-tuning problem not resolved - What physical field causes inflation? - What ends inflation? Existing alternatives to GR: • Dark energy • Use exotic fields • Are more complicated • Dark matter • Do not address all problems • Matter-antimatter asymmetry (usually 1, sometimes 2) Einstein-Cartan-Sciama-Kibble theory
Spacetime with gravitational torsion
This talk: Big-bang singularity, inflation and dark energy problems all naturally solved by torsion Affine connection
• Vectors & tensors – under coordinate transformations behave like differentials and gradients & their products.
• Differentiation of vectors in curved spacetime requires subtracting two vectors at two infinitesimally separated points with different transformation properties.
• Parallel transport brings one vector to the origin of the other, so that their difference would make sense.
dx
A δA Affine connection E = mc2
Curvature and torsion
Calculus in curved spacetime requires geometrical structure: affine connection
Covariant derivative of a vector
Two tensors constructed from affine connection:
• Torsion tensor – antisymmetric part of connection
É. Cartan (1921)
Contortion tensor E = mc2
Theories of spacetime
Special Relativity – flat spacetime (no affine connection) Dynamical variables: matter fields
General Relativity – (curvature, no torsion) Dynamical variables: matter fields + metric tensor
Degrees Connection restricted to be symmetric – ad hoc of freedom (equivalence principle)
ECSK gravity (simplest theory with curvature & torsion) Dynamical variables: matter fields + metric + torsion E = mc2
ECSK gravity T. W. B. Kibble, J. Math. Phys. 2, 212 (1961) D. W. Sciama, Rev. Mod. Phys. 36, 463 (1964)
Riemann-Cartan spacetime – metricity
→
Christoffel symbols of metric Matter Lagrangian density
Total Lagrangian density like in GR:
Two tensors describing matter:
• Energy-momentum tensor
• Spin tensor E = mc2
ECSK gravity
Curvature tensor = Riemann tensor + tensor quadratic in torsion + total derivative
Stationarity of action under → Einstein equations
Stationarity of action under → Cartan equations
• Torsion is proportional to spin density • Contributions to energy-momentum from spin are quadratic Dirac spinors with torsion
Simplest case: minimal coupling
Dirac Lagrangian density (natural units)
Dirac equation
Covariant derivative of a spinor GR covariant derivative of a spinor
F. W. Hehl, P. von der Heyde, G. D. Kerlick & J. M. Nester, Rev. Mod. Phys. 48, 393 (1976) Dirac spinors with torsion
Spin tensor is completely antisymmetric
Torsion and contortion tensors are also antisymmetric
LHS of Einstein equations
comoving frame
Fermion number density E = mc2
ECSK gravity
Torsion significant when Uik » Tik (at Cartan density)
45 -3 For fermionic matter ½C > 10 kg m >> nuclear density
Other existing fields do not generate torsion
• Gravitational effects of torsion are negligible even for neutron stars (ECSK passes all tests of GR)
• Torsion vanishes in vacuum → ECSK reduces to GR
• Torsion is significant in very early Universe and black holes
Imposing symmetric connection is unnecessary ECSK has less assumptions than GR E = mc2
Cosmology with torsion
Spin corrections to energy-momentum act like a perfect fluid
Friedman equations for a homogeneous and isotropic Universe:
Statistical physics in early Universe (neglect k) E = mc2
Cosmology with torsion NP, Phys. Rev. D 85, 107502 (2012)
Scale factor vs. temperature
Solution reference values
torsion correction
Singularity avoided E = mc2
Cosmology with torsion
Temperature vs. time
→
Can be integrated parametrically
t ≤ 0
t ≥ 0 E = mc2
Cosmology with torsion
Temperature vs. time
Time scale of torsion era
Cusp-like bounce E = mc2
Nonsingular big bounce instead of big bang
Scale factor vs. time NP, Phys. Rev. D 85, 107502 (2012)
Big bounce E = mc2
Nonsingular big bounce instead of big bang
Scale factor vs. time
slope discontinuity
Big bounce E = mc2
Nonsingular big bounce
Scale factor vs. time
? Mass generation
Big bounce E = mc2
Nonsingular big bounce
Singularity theorems?
Spinor-torsion coupling enhances strong energy condition
Expansion scalar (decreasing with time) in Raychaudhuri equation
is discontinuous at the bounce, preventing it from decreasing to (reaching a singularity) E = mc2
Torsion as alternative to inflation
For a closed Universe (k = 1):
Velocity of the antipode relative to the origin
At the bounce
Density parameter Current values (WMAP)
NP, Phys. Rev. D 85, 107502 (2012) E = mc2
Torsion as alternative to inflation
Big bounce:
Minimum scale factor
Horizon problem solved Number of causally disconnected volumes
Flatness problem solved No free parameters
Cosmological perturbations – in progress E = mc2
Theories of spacetime
General Relativity Dynamical variables: matter fields + metric tensor
ECSK gravity Dynamical variables: matter fields + metric tensor + torsion
Purely affine gravity (A. Eddington 1922, A. Einstein 1923, E. Schrödinger 1950) Dynamical variables: matter fields + affine connection
• Metric tensor is constructed from matter Lagrangian & curvature • Field equations in vacuum generate cosmological constant • Field equations with matter are more complicated and differ from (physical) metric solutions E = mc2
Affine gravity
Similar to gauge theories of other fundamental forces:
• Affine connection (dynamical variable in affine gravity) generalizes an ordinary derivative to a coordinate-covariant derivative
• Gauge potentials (dynamical variables in gauge theories) generalize an ordinary derivative to gauge-invariant derivatives
E = mc2
Affine gravity
Dynamical Lagrangian must contain derivatives of connection
Simplest gravitational Lagrangian: linear in derivatives
→ linear in Ricci tensor (like in GR and ECSK) and contracted with an algebraic tensor constructed from connection (from torsion)
Other Lagrangians based on
NP, ArXiv:1203.0294 are unphysical E = mc2
Affine gravity
Stationarity of action under → field equations
Variation can be split into and
For vacuum:
→
Christoffel symbols of tensor k
Gravitational Lagrangian becomes
Ricci tensor of tensor k Cosmological-like term E = mc2
Cosmological constant from torsion
Defining
Λ sets length scale of affine connection gives the Einstein-Hilbert action with cosmological constant
Affine length scale Λ becomes cosmological constant
Only configurations with are physical c, Λ, G – fundamental constants of classical physics (set units) Planck units set by h, c, G – their relation to Λ still unknown
NP, ArXiv:1203.0294 E = mc2
Cosmological constant from torsion
The metric in the matter Lagrangian must also be replaced by
For ordinary matter (Dirac spinors, known gauge fields): same gravitational Lagrangian
Total action
sets mass units becomes the EH action with matter and Λ E = mc2
Cosmological constant from torsion
If fields depend on torsion only through (spinors do not):
→
Equations in the presence of spinors – in progress
Expected to reproduce or slightly modify ECSK with Λ
1/2 2 These modifications may contain Λ c ~ aMOND → dark matter
Summary
Torsion in the ECSK theory of gravity:
• Averts the big-bang singularity, replacing it by a nonsingular, cusp-like big bounce
• Solves the flatness and horizon problems without inflation
Torsion in the simplest affine theory of gravity:
• Gives field equations with a cosmological constant
No free parameters